To introduce the quotient geometry,
assume \({\cal X}\)  is represented in the TT format (REF )
with core tensors \(\bar{{\cal X}}:=\left\lbrace {\cal X}^1,\cdots ,{\cal X}^d\right\rbrace \) . It is known that the condition \(\operatorname{\mathrm {rank}}_{\operatorname{TT}}\left({\cal X}\right)= \mathbf {r}\)  is equivalent to
\(\operatorname{\mathrm {rank}}\left(L\left({\cal X}^k\right)\right)= r_k\)  and \(\operatorname{\mathrm {rank}}\left(R\left({\cal X}^k\right)\right)= r_{k-1}\) , \(k = 1,\cdots ,d\)  [1]}, [2]}. We denote by \(\overline{{\cal M}}_{\mathbf {r}} := \mathbb {R}_{\ast }^{r_0\times n_1\times r_1}\times \cdots \times \mathbb {R}_{\ast }^{r_{d-1}\times n_d\times r_d}\)  the set of tensors with the following rank constraint: for any \({\cal X}^k\in \mathbb {R}_{\ast }^{r_{k-1}\times n_k\times r_{k}}\) , \(\operatorname{\mathrm {rank}}\left(L\left({\cal X}^k\right)\right)= r_k\) , and \(\operatorname{\mathrm {rank}}\left(R\left({\cal X}^k\right)\right)= r_{k-1}\) , \(k = 1,\cdots ,d\) . Let the mapping \(\phi \)  be
\(\phi : \overline{{\cal M}}_{\mathbf {r}}\rightarrow {\cal M}_{\mathbf {r}} : \bar{{\cal X}} \rightarrow \phi \left(\bar{{\cal X}}\right), \hbox{ such that } \phi \left(\bar{{\cal X}}\right)\left(i_1,\cdots ,i_d\right)= {\cal X}^1\left(i_1\right){\cal X}^2\left(i_2\right)\cdots {\cal X}^d\left(i_d\right).\) 
